MV-algebras, infinite dimensional polyhedra, and natural dualities

TL;DR

This paper links the duality theory of MV-algebras with natural dualities, simplifying the construction and providing geometric descriptions, while extending results on semisimple tensor products and polyhedral MV-algebras.

Contribution

It simplifies the duality construction for MV-algebras, introduces geometric dual maps, and extends characterizations of semisimple and polyhedral MV-algebras beyond finitely generated cases.

Findings

Simplified duality construction for MV-algebras.

Provided geometric descriptions of dual maps.

Extended characterizations of semisimple and polyhedral MV-algebras.

Abstract

We connect the dual adjunction between MV-algebras and Tychonoff spaces with the general theory of natural dualities, and provide a number of applications. In doing so, we simplify the aforementioned construction by observing that there is no need of using presentations of MV-algebras in order to obtain the adjunction. We also provide a description of the dual maps that is intrinsically geometric, and thus avoids the syntactic notion of definable map. Finally, we apply these results to better explain the relation between semisimple tensor products and coproducts of MV-algebras, and we extend beyond the finitely generated case the characterisations of strongly semisimple and polyhedral MV-algebras.